optimal control theory
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Optimal Control Theory. Prof .P.L.H .Vara Prasad. Dept of Instrument Technology Andhra university college of Engineering. Overview of Presentation. What is control system Darwin theory Open and closed loops Stages of Developments of control systems Mathematical modeling - PowerPoint PPT PresentationTRANSCRIPT
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Optimal Control Theory
Dept of Instrument TechnologyAndhra university college of Engineering
Prof .P.L.H .Vara Prasad
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Overview of Presentation
What is control system Darwin theory Open and closed loops Stages of Developments of control systems Mathematical modeling Stability analysis
Dept of Inst TechnologyAndhra university college of Engineering
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What is a control system ?
A control system is a device or set of devices to manage, command, direct or regulate the behavior of other devices or systems.
Dept of Inst TechnologyAndhra university college of Engineering
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Darwin (1805)Feedback over long time periodsis responsible for the evolution of species.
Dept of Inst TechnologyAndhra university college of Engineering
vito volterra - Balance between two populations of fish(1860-1940)
Norbert wiener - positive and negative feed back in biology (1885-1964)
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Open loop & closed loop
“… if every instrument could accomplish its own work, obeying or anticipating the will of others … if the shuttle weaved and the pick touched the lyre without a hand to guide them, chief workmen would not need servants, nor masters slaves.”
Hall (1907) : Law of supply and demand must distrait fluctuations
Any control system- Letting is to fluctuate and try to find the dynamics.
Dept of Inst TechnologyAndhra university college of Engineering
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Open loop Accuracy depends
on calibration. Simple. Less stable. Presence of non-
linearities cause malfunctions
Open loop Accuracy depends
on calibration. Simple. Less stable. Presence of non-
linearities cause malfunctions
Closed loop
Due to feed back
Complex
More stable
Effect of non-linearity can be minimized by selection of proper reference signal and feed back components
Closed loop
Due to feed back
Complex
More stable
Effect of non-linearity can be minimized by selection of proper reference signal and feed back components
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Effects of feedback
System dynamics normal improved Time constant 1/a 1/(a+k) Effect of disturbance
Direct -1/g(s)h(s) reduced
Gain is high low gain G/(1+GH)
If GH= -1 , gain = infinity
Selection of GH is more important in finding stable
low Band width high band width
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Robot using pattern- recognition process
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Temperature control system
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Analogous systems
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Mathematical model of gyro
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Mathematical modeling of physical systems
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Stages of Developments of control systems
Dept of Inst TechnologyAndhra university college of Engineering
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Example of 2nd order system
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optimization
Maximize the profit or to minimize the cost dynamic programming .
Non linear optimal control
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Nature of response -poles
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Unit step response of a control system
Dept of Inst TechnologyAndhra university college of Engineering
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Steady state errors for various types of instruments
Dept of Inst TechnologyAndhra university college of Engineering
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For Higher order systems Rouths –Hurwitz stability criterion & its application
Dept of Inst TechnologyAndhra university college of Engineering
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Locus of the Roots of Characteristic Equation
Dept of Inst TechnologyAndhra university college of Engineering
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Root Contour
Dept of Inst TechnologyAndhra university college of Engineering
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Performance Indices
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Frequency response characteristics- Polar plots
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Bode plots
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Phase & gain margins
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Nyquist plots
First order system Second order system Third order system
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Nyquist stability
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Limitations of Conventional Control Theory
Applicable only to linear time invariant systems. Single input and single output systems Don’t apply to the design of optimal control systems Complex Frequency domain approach
Trial error basisNot applicable to all types of in putsDon't include initial conditions
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State Space Analysis of Control Systems
Definitions of State Systems Representation of systems Eigen values of a Matrix Solutions of Time Invariant System State Transition Matrix
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Definitions
State – smallest set of variables that determines the behavior of system
State variables – smallest set of variables that determine the state of the dynamic system
State vector – N state variables forming the components of vector
Sate space – N dimensional space whose axis are state variables
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State space representation
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State Space Representation
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Solutions of Time Invariant System Solution of Vector Matrix Differential
Equation X|= Ax (for Homogenous System) is given by
X(t) = eAt X(0) (1)
Ø(t) = eAt = L -1 [ (sI-A)-1 ] (2)
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Solutions of Time Invariant System…(Cont’d)
Solution of Vector Matrix Differential Equation X|= Ax+Bu
(for Non- Homogenous System) is given by
X(t) = eAt X(0) + ∫t0
e ^{A(t - T)} * Bu(T) dT
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Optimal Control Systems Criteria
Selection of Performance Index Design for Optimal Control within
constraints
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Performance Indices
Magnitudes of steady state errors Types of systems Dynamic error coefficients Error performance indexes
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Optimization of Control System State Equation and Output Equation Control Vector Constraints of the Problem System Parameters Questions regarding the existence of
Optimal control
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Controllability
A system is Controllable at time t(0) if it is possible by means of an unconstrained control vector to transfer the System from any initial state Xt(0) to any other state in a finite interval of time.
Consider X| = Ax+Bu then system is completely state controllable if the rank of the Matrix
[ B | AB | …….An-1B ] be n.
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Observability A system is said to be observable at time t(0) if,
with the system in state Xt(0) it is possible to determine the state from the observation of output over a finite interval of time.
Consider X| = Ax+Bu, Y=Cox then system is completely state observable if rank of N * M matrix [C* | A*C* | …… (A*)n-1 C*] is of rank n .
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Liapunov Stability Analysis
Phase plane analysis and describing function methods – applicable for Non-linear systems
Applicable to first and second order systems Liapunov Stability Analysis is suitable for
Non-linear and|or Time varying State Equations
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Stability in the Sense of Liapunov
Stable Equilibrium state Asymptotically Stable Unstable state
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Liapunov main stability theorem
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Thank you